Let (X, {\cal F}, l) be the unit circle \Bbb S1 = \{z \in \Bbb C : |z| = 1\} with the usual s-algebra {\cal F} of Lebesgue measurable subsets and the normalized Lebesgue measure l. Consider a sequence nn: \Bbb N \ra \Bbb R, \;\; nn(k) \geq 0, \;\; S\inftyk=1 nn(k) = 1. For any measure-preserving t : X \ra X, this sequence induces a sequence (Tn)\infty1 of bounded, linear operators on Lp(X), \;\; 1 \leq p \leq \infty, by defining \[ Tn f = \sum\inftyk=1 nn(k) \; f \circ tk, \quad n = 1, 2, \ldots . \] We shall prove that under suitable conditions imposed on t and (nn)\infty1, there exists a large collection of measurable characteristic functions f for which \lim \supn \ra \infty Tn f - \lim \infn \ra \infty Tn f = 1 a.e on X.

Let (X, {\cal F}, l) be the unit circle \Bbb S1 = \{z \in \Bbb C : |z| = 1\} with the usual s-algebra {\cal F} of Lebesgue measurable subsets and the normalized Lebesgue measure l. Consider a sequence nn: \Bbb N \ra \Bbb R, \;\; nn(k) \geq 0, \;\; S\inftyk=1 nn(k) = 1. For any measure-preserving t : X \ra X, this sequence induces a sequence (Tn)\infty1 of bounded, linear operators on Lp(X), \;\; 1 \leq p \leq \infty, by defining \[ Tn f = \sum\inftyk=1 nn(k) \; f \circ tk, \quad n = 1, 2, \ldots . \] We shall prove that under suitable conditions imposed on t and (nn)\infty1, there exists a large collection of measurable characteristic functions f for which \lim \supn \ra \infty Tn f - \lim \infn \ra \infty Tn f = 1 a.e on X.